The pricing of an option of an asset is a fundamental problem of significant practical importance in today""s financial markets. In 1973, a mathematician, Fischer Black, and an economist, Myron Scholes, devised one of the first mathematically accepted approaches for pricing what is known as a xe2x80x9cEuropeanxe2x80x9d option, which are options that can.only be exercised at its expiration date. What has become known as the Black-Scholes option formula was described first in xe2x80x9cThe pricing of options and corporate liabilities,xe2x80x9d Journal of Political Economy 81 (1973). The Black-Scholes option formula is presently of widespread use in financial markets all over the world. The price of such an option can be found by solving the Black-Scholes equation with the initial condition at expiration (i.e., the payoff of the option). The Black-Scholes equation is a reverse diffusion equation with parameters determined by the statistical characteristics of involved stocks and currencies such as risk free interest rate, holding cost or expected dividends, and volatility.
As an example, the Black-Scholes formula for the theoretical price of a vanilla European call option is:
C(S, t)=SN(d1)xe2x88x92Eexe2x88x92r(Txe2x88x92t)N(d2),xe2x80x83xe2x80x83(1)
where the notation is fairly standard, as described by P. Wilmott, J. N. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford (1993).
Unlike a European option, an xe2x80x9cAmericanxe2x80x9d option gives the owner of the option the right of exercising the option before its expiration date. In recent years, the American option has become more prevalent than the European option. Due to the additional feature of early exercise, the pricing of an American option is generally considered to be more difficult than the pricing of a European option, especially when one considers xe2x80x9cexoticxe2x80x9d options which are variations and refinements of a basic American option. From a mathematical point of view, part of this difficulty is due to the price of American options obeying the Black-Scholes equation only in the region where it is statistically appropriate to hold the option rather than to exercise it immediately.
Therefore, in the case of American options, the above formula (1) and its analogs are no longer valid. In fact, as shown in P. Jaillet, D. Lamberton, and B. Lapeyre, xe2x80x9cVariational inequalities and the pricing of American options,xe2x80x9d Acta Applicandae Mathematicae 21 (1990) 263-289, a rigorous mathematical model for pricing an American option is an infinite-dimensional free boundary problem. As such, there is in general no explicit formula or finite procedure for computing the exact price of an American option. As a result, various mathematical models have been devised in an attempt to approximate the price of such options.
However, the option prices computed from a mathematical model are of a theoretical nature. In computing these prices, various inputs are fed into the model and an algorithm produces an answer. In practice, the computed prices may not be consistent with the observed market prices, e.g., the prices on the trading floor. Ideally, these two sets of prices should coincide. However, such a result is difficult, if not impossible, using known models. Two principal reasons for this are: (i) the assumptions that lead to the construction of the mathematical model may not be realistic; and (ii) the inputs to the model are not correct.
Knowing the correct inputs is crucial to the success of any pricing model. Ideally the two sets of option prices, computed and observed, are within an acceptable range of one another. In general, the problem of computing a proper set of inputs to a forward pricing model so that the computed outputs obey certain prescribed criteria is called an xe2x80x9cinversexe2x80x9d pricing problem. One such input which is crucial to the forward pricing model of American options is the implied volatility of an option. The volatility of an asset is an important input to an option pricing model and it is also an input that is most difficult to estimate. Black and Scholes assumed this parameter to be a constant when deriving their famous formula for the theoretical price of a vanilla European call option. In the above equation (1), for example, the two constants d1 and d2 contain the volatility parameter "sgr". As it is now well known, this parameter is, in general, not a constant; indeed it is a highly complicated function of several deterministic and random factors.
Previous approaches for dealing with this difficult problem of unknown volatility are numerous and include: (i) statistical estimation methods based on historical data (such as J. Hull, Options, Futures, and Other Derivative Securities, Second Edition, Prentice Hall, New Jersey (1989), Section 10.4 and R. Gibson, Option Valuation: Analyzing and Pricing Standardized Option Contracts, McGraw-Hill, New York (1991), Section 1; (ii) mathematical models of stochastic volatilities such as those in J. Hull and A. White, xe2x80x9cThe pricing of options on assets with stochastic volatilities,xe2x80x9d The Journal of Finance 42 (1987) 281-300; H. Johnson and D. Shanno, xe2x80x9cOption pricing when the variance is changing,xe2x80x9d Journal of Financial and Quantitative Analysis 22 (1987) 143-151; and (iii) implied volatilities based on observed option prices (suggested originally by H. A. Latant and R. J. Rendleman, xe2x80x9cStandard deviations of stock price ratios implied in option prices,xe2x80x9d The Journal of Finance 31 (1976) 369-381 and empirically tested by S. Beckers, xe2x80x9cStandard deviation implied in option prices as predictors of future stock price volatilityxe2x80x9d Journal of Banking and Finance 5 (1981) 363-381).
The problem of computing implied volatilities of (European or American) options is an instance of an inverse problem that is the counterpart of the forward problem of pricing these options. Specifically, in the forward option pricing problem, a constant volatility parameter, along with other constants, such as the interest rate and asset dividend, is taken as an input to a mathematical model that produces the theoretical or modeled option prices. By using an incorrect volatility parameter in the forward pricing model, the computed option price is bound to deviate, often substantially, from the option price observed on the trading floor.
An apparatus for and method of determining the price of financial derivatives such as options. One preferred embodiment of the invention employs a discretized partial differential linear complementarity problem (PDLCP) based system to determine the forward pricing of financial instruments such as vanilla American options. In this embodiment, an optimization problem in the form of a mathematical program with equilibrium constraints (MPEC) is implemented to derive implied volatilities of the assets underlying the subject derivatives. The implied volatilities thus derived are used as inputs in the PDLCP-based system to accurately determine the forward pricing of the subject derivatives.